Immiscibility of ionic melts: a simple model with charge differences

This work is devoted to the analysis of the immiscibility mechanism and the peculiarities of its manifestation in the case of mixtures of classical electrolytes. This mechanism should be deduced from the differences in the potential energy of the ions constituting the components of the mixture with respect to their surroundings. Since electrostatic interactions are shielded at a large distance from the central ion in any electrolytes, it is important for the considered mechanism what contribution to the concentration dependence of the chemical potential of a component is given by one or another sort of ions. In this paper we consider a simplified model of a binary solution in which the interaction of cations and anions in each of the ionic liquids is approximated by the model of charged hard spheres, i.e., they are considered as primitive electrolytes. Since the problem of liquid-phase immiscibility cannot be considered without taking into account the finite sizes of ions, it is necessary, firstly, to choose at least the full version of the Debye-Hückel model, and, secondly, to take into account the direct contribution of excluded volume effects or hard-sphere repulsion, for which a van der Waals-type model can be used. As a result, the reasoning about the concentration dependence of the density in a liquid-phase system and the equation of state that allows us to find it become key for describing the features of the miscibility gap. The theoretical analysis of the immiscibility problem can be carried out by considering that a cation and an anion belonging to one of the components of a binary mixture possess the same value of ionic radius and equal but opposite charge, while differing in their values for the other component of the solution. Thus, a binary restricted primitive model (RPM) is formulated to consider the effects of charge differences on the miscibility gap. In the present work, analytical expressions describing the position of the critical point in the asymptotic limit of small charge differences are derived in detail. It is shown that the critical temperature is proportional to the fourth degree of the charge mismatch, and the shift of the critical composition from equimolar occurs towards the component with smaller charge values. The latter result seems to be quite general, describing the preference in solubility of salts, which have larger charge values, in ionic melts with smaller charges on cations and anions.